A note on the hyperarithmetical hierarchy

The hyperarithmetical hierarchy assigns a degree of unsolvability h γ to each constructive ordinal γ . This assignment has the properties that h 0 is the recursive degree and h γ +1 is the jump h ′ γ of h γ . For a limit ordinal λ < ω 1 it is not so easy to define h γ . The original definitions used systems of notations for ordinals, see Spector [6]. There are also later notation-free definitions due to Shoenfield (unpublished) and to Hensel and Putnam [4].